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Teaching Math: A Video Library, 9-12

Exploring Functions

Content Standards
Functions, Algebra, Geometry from a Synthetic Perspective
Process Standards
Reasoning, Problem Solving, Communication, Connections
School: San Juan High School; 1,250 students
Location: Citrus Heights, California
Teacher: Jeanne Shimizu–Yost
Years Teaching: 14
Students in Classroom: 22
Grade: 10 – 12
Exploring Functions
Video Overview
Ms. Shimizu–Yost asks her students to work in groups to solve two equations using more than one method—such as graphing, algebra, or geometry. To solve the equations, the students use graphing calculators and individual resource sheets created by each student, called yellow pages. Each group records its observations and solutions on poster board. Then Ms. Shimizu–Yost reconvenes the class, and each group presents its work to the class. At the end of the lesson, the whole class discusses the solutions.
The Class Challenge
Students solve each of the equations below for x.
2^x = 25
2^x = x+3

Course and Curriculum
This elective Algebra 2 course is the fourth course in the school mathematics program and the third course in the College Preparatory Mathematics (CPM) curriculum. Prior to this course, students took Math A, Algebra I, and Geometry. The course emphasizes functions; problem solving, including algebraic and geometric approaches; functions; and understanding data represented algebraically, geometrically, and narratively.
Prior to this lesson, students studied inverse and exponential functions. Ms. Shimizu–Yost created this lesson to assess how her students would apply their understanding of exponential functions, logarithmic functions, and problem–solving techniques to a new situation.
Ms. Shimizu–Yost uses the yellow pages (resource sheets created by students) to develop students' study, self–assessment, and independent thinking skills. In groups, which change for each unit, students learn new concepts and review material. At the end of each unit, Ms. Shimizu–Yost gives students a group test and a smaller individual test.

A Pre–Viewing Exploration for Teacher Workshops
Multiple Representations of Exponential Functions
Objective: To investigate algebraic, geometric, and estimated solutions to exponential functions.
Materials: Graph paper and a graphing calculator for each teacher.
Activity
Solve this problem individually.
Explore solutions to 3^x = 40.

1 What is your estimate of the value of x?

2 Solve this problem algebraically. Use logarithms by taking the log of each side of the equation. This gives:
     log 3^x=log 40
x log 3=log 40
x=(log 40)/(log 3)
Now use your calculator to find the value of x.

3 To solve the problem geometrically, convert the equation to a function to find the value of x when the function is equal to 0.
3x = 40
0 = 40 – 3^x
y = 40 – 3^x
3a On graph paper, graph the function using at least 5 values for x from –2 to 4. What is the approximate value of x when y equals 0? Graph on a calculator to find the exact value.

3b Another way to represent the solution graphically is to graph each side of the equation 3^x = 40. Write each side as a function and graph each function on graph paper or a calculator. Where do the two graphs intersect?
Exploring Content
1 How do the estimated and algebraic solutions compare?
2 In the geometric solutions, how do the graphs using two different methods compare?
3 What represents the solution in the graphs of each method?
4 How would the solution change if the equation were changed to 2^x = 2x?
Exploring Teaching Issues
1 How would you respond if your students found two different solutions when they used two different methods?
2 How might using multiple approaches deepen your students' understanding of this or similar equations?
3 Identify one type of equation in your curriculum and the confusion students encounter when solving it.

Exploration Answers
Activity
1 Between 3.0 and 3.9.
2 The value of x is 3.357762781.
3a 3.35
3b 3.35

Exploring Content
1 Solutions should be close. A good estimate is 3.1; the actual answer is 3.357762781.
2 Graphs appear different, but the interpretation of the graphs yield the same solution. The estimated solution is between 3.0 and 3.9, and the algebraic solution is 3.357762781. The first graph represents the converted function, and the second graph has two intersecting graphs.
3 In the first graph, it is the x–intercept; in the second graph, it is the point of intersection of the graphs.
4 The graphs would intersect in two points.

Topics for Discussion
Consider the use of yellow pages.

How did yellow pages resource sheets encourage communication?
How were yellow pages used to support skill development and conceptual understanding?
One student wrote the following first two steps in her solution to 2^x = 25: (log) 2^x = 25 (log); log 2^x = log 25. What is her conceptual misunderstanding, and what would you expect her to write in her yellow pages as a result? How would you help her understand what log means?
Discuss the advantages of solving a problem using multiple methods.

How did using multiple methods help students develop their confidence in problem solving?
What criteria did students use to validate their strategies?
Compare and discuss two graphical approaches to solving 2^x = x + 3.
Why is it important for students to use a variety of methods for solving different algebraic equations and for examining patterns of a variety of algebraic functions?
Why do you think students did not use numerical methods (for example, by using a table to organize the numbers and look for patterns)?
What important mathematical topics and tasks are supported when multiple methods are used to solve problems?
Why do you think Ms. Shimizu–Yost had her students do this lesson in small groups? What other approaches could students use to explore multiple methods?
What connections within mathematics were considered?

When Ms. Shimizu–Yost asked her students why they concluded that 2^x = x + 3 cannot be solved algebraically, one student explained that they could not get rid of the x by the algebraic method because they do not know how yet. How would you use this student's response to inform your teaching?
Cite examples of how students communicated mathematical ideas, reflected upon and clarified their thinking, and expressed their mathematical ideas.
What was revealed through students' summaries of their solutions during their group explorations and preparation for presentations?
What method would you use to solve equations that have an exponential and linear component? Where could you use that method in your current algebra or advanced algebra curriculum?